Abstract

We consider the following distributed service model: jobs with unit mean, exponentially distributed, and independent processing times arrive as a Poisson process of rate λ N, with 0<λ<1, and are immediately dispatched to one of several queues associated with N identical servers with unit processing rate. We assume that the dispatching decisions are made by a central dispatcher endowed with a finite memory, and with the ability to exchange messages with the servers. We study the fundamental resource requirements (memory bits and message exchange rate), in order to drive the expected steady-state queueing delay of a typical job to zero, as N increases. We propose a certain policy and establish (using a fluid limit approach) that it drives the delay to zero when either (i) the message rate grows superlinearly with N, or (ii) the memory grows superlogarithmically with N. Moreover, we show that any policy that has a certain symmetry property, and for which neither condition (i) or (ii) holds, results in an expected queueing delay which is bounded away from zero.

Highlights

  • This paper addresses the tradeoffs between various measures of performance in the context of large scale queueing systems

  • The main objective of this paper is to find necessary and sufficient conditions on the amount of resources available to a central dispatcher, in order to achieve a vanishing queueing delay as the system size increases

  • This is done by defining a unified framework for a broad class of dispatching policies and by proving two separate results: First, we show that when we have a limited amount of memory and a modest budget of messages per unit of time, all dispatching policies result in queueing delay uniformly bounded away from zero

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Summary

INTRODUCTION

This paper addresses the tradeoffs between various measures of performance (delay, message rate and memory) in the context of large scale queueing systems. At the other extreme, incoming jobs can be sent to a shortest queue, or to a server with the smallest workload The latter policies have very good performance (small queueing delay), but rely on a substantial information exchange overhead. Many intermediate policies have been explored in the literature, which achieve different performance levels while using varying degrees of resources such as messages and memory (e.g., the power-of-d-choices [11, 15] or the PULL algorithm [14]). Such policies have been individually analyzed, and the merits of each have been pointed out.

Our contribution
Outline of the paper
NOTATION
MODEL AND MAIN RESULTS
Modeling assumptions
Policy description
System state representation
Fluid model
Properties of the fluid solutions
Approximation theorems
Unified framework for policies
Lower bound for queueing delay
COMPARISON WITH OTHER POLICIES IN THE LITERATURE
Policies based on queue lengths
Pull-based policies
Uniqueness of solutions
Asymptotic stability of the equilibrium
INTERCHANGE OF LIMITS
PROOF OF THE NEGATIVE RESULT
The memory is updated according to
CONCLUSIONS AND FUTURE WORK
Full Text
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